Semidirect product of a Lie algebra and a module ($\mathfrak{g}⋉V$)

For a Lie algebra representation of $\mathfrak{g}$ on $V$, the semidirect product $\mathfrak{g}⋉V$ is the vector space $\mathfrak{g}\oplus V$ with bracketThe subspace $V$ is an abelian ideal of a Lie algebra.

For a finite-dimensional representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$,In particular, $V$ lies in the radical of the Killing form.